I will discuss how to construct a derivation from a quantum Markov semigroup (after Cipriani-Sauvageot). This derivation can be then used to obtain strong solidity results, via Hilbert-Schmidt estimates for certain maps associated with the semigroup. I also plan to describe applications to the Akemann-Ostrand property, which usually rely on nuclearity of certain C*-algebras. New results are obtained for q-Gaussian algebras, but also many known examples are recovered.

This is joint work with Martijn Caspers and Yusuke Isono.